PILOTBEAR: it's interesting that you define slips that way because the canadian definitions are opposite what you just posted.
In Canada, what you call the slip depends entirely on your relation with the runway.
In a forward slip, you are "on the localizer" and tracking to the runway, but your longitudinal axis is misaligned. Used to lose altitude.
In a side slip, you are also either tracking in line with the runway with one wing low (only possible in x-wind)
Here's where it gets funny:
We also call it a sideslip when your longitudinal axis is in line with the runway, but you are off the centerline, and tracking to the side to get on the centerline. In this case, your track on the ground is off at an angle, as in a forward slip. The only difference is that the purpose is to change the direction of flight instead of changing the rate of descent.
This is out of the Transport Canada Flight Training Manual (which I loathe)
These definitions are tedious, stupid, and meaningless, since the aerodynamics are identical in all cases if you take away the ground. My biggest problem is that many instructors i've spoken with don't realize this fact, and often, they don't even realize that in a side slip, your rate of descent increases. I would rather term them all simply "slips", and only differentiate when teaching the slipping turn, as it is aerodynamically different.
Of course, I'm in Canada, so I will have to mention the difference in the terms, but I will stress that is is the SAME maneuvre, the only difference is the purpose.